While examples with a known ground truth are useful to judge quantities such as the epistemic uncertainty or (REF ), in practice the noise-free ground truth will usually be unknown, even for the training data. As two final examples we will therefore study two real world examples, where no ground truth is available and which are available as open datasets. We consider the prediction of the quality of Portuguese red "Vinho Verde" [1]}, that contains for each of the 1599 studied wines a quantity quantifying the quality and 11 chemical properties, as well as the Boston Housing dataset [2]}. For the first dataset all concentrations were transformed by a logarithm and the inputs and output were normalized to have mean 0 and standard deviation 1. A subset of 20% was split off for testing purposes. On the remaining a fully connected neural network with three hidden layers (with 200, 100 and 50 neurons) was trained for 400 epochs with a batch size of 16, \(\mbox{\small $(2N\lambda _\theta ^2)^{-1}$}=10^{-9}\)  and various choices of Deming factors \(\delta \) . On the second dataset, the disputed [3]} 12th column was removed, inputs and output were normalized and a subset of 20% was split off for testing. On the remainder a fully connected neural network with three hidden layers (200, 100 and 50 neurons) was trained for 1000 epochs with a batch size of 16 and a strong regularization \(\mbox{\small $(2N\lambda _\theta ^2)^{-1}$}=10^{-2}\)  (to prevent overfitting). Figure REF  shows the average of the RMSE, together with its standard error, on the test set depending on the used \(\delta \)  for 20 training runs with different randoms seeds for both datasets. For the non-EiV model the area of one standard error around the mean is depicted in gray. We saw already in Table REF  that the RMSE cannot be used as a sole measure to judge the quality of the used model: a slightly worse RMSE can come with a much better uncertainty coverage. However, studying the RMSE can help to rule out too large values for \(\delta \)  that lead to a bad fit and overprudent small \(\delta \) . From Figure REF  a good choice for the wine quality dataset might be \(\delta =0.6\)  with a relatively small RMSE. The “symmetric” choice \(\delta =1\)  (with \(\sigma _x=\sigma _y\) ) on the other hand yields only a slightly worse RMSE as the non-EiV model. In fact, as we can see in Figure REF  the errors that both models make for \(\delta =1\)  are almost identical. The uncertainty, on the other hand, depicted in Figure REF , is increased for the EiV model. This is identical to the behavior observed for the toy examples in Figure REF  and REF . As we have no ground truth available for this dataset a coverage as in Table REF  cannot be computed.
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